Power of a Power

≈ 25 min

Raising a power to another power means you multiply the exponents:

(a^m)^n = a^{m \times n}

Why? $(2^3)^2$ means $2^3$ multiplied by itself:

(2^3)^2 = 2^3 \times 2^3 = 2^{3+3} = 2^6

Two lots of three factors is $3 \times 2 = 6$ factors.

A related law: a product inside a bracket shares the exponent among each factor:

(ab)^n = a^n b^n \qquad\text{so}\qquad (2x)^3 = 2^3 x^3 = 8x^3

Worked example. Simplify $(k^4)^3$ .

Multiply the exponents: $k^{4 \times 3} = k^{12}$ .

Evaluate $(3^2)^2$ .

Simplify $(x^2)^5$ .

$x^7$

$x^{10}$

$x^{25}$

$x^3$

Write $(2^3)^4$ as a single power, in the form base^exponent.

Simplify $(2x)^3$ . Write your answer in the form ax^n.